Theorems · Theorem · linear algebra
rank_range_le
∀ {R : Type u} {M M₁ : Type v} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M]
[inst_3 : AddCommMonoid M₁] [inst_4 : Module R M₁] (f : M →ₗ[R] M₁), Module.rank R ↥f.range ≤ Module.rank R M- Defined in
- Mathlib.LinearAlgebra.Dimension.Basic
- Cited by
- 2 results in Mathlib
- Foundations
- Depth 86 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites11
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Modulestatement and proof · cited by 20,661
- RingHom.idstatement and proof · cited by 18,349
- Semiringstatement and proof · cited by 13,802
- AddCommMonoidstatement and proof · cited by 12,281
- LinearMapstatement and proof · cited by 10,215
- Submodulestatement · cited by 7,192
- Cardinalstatement · cited by 2,598
- LinearMap.rangestatement and proof · cited by 893
- Module.rankstatement and proof · cited by 496
- Cardinal.lift_idproof · cited by 163
- lift_rank_range_leproof · cited by 7
Cited by2
Results whose statement or proof uses this declaration.
- LinearMap.rank_le_of_surjectiveproof · cited by 2
- LinearMap.rank_le_domainproof · cited by 1